algorithms

9.2 Approximation algorithms
In an optimization problem we are given an instance I and are asked to find the optimum
solution—the one with the maximum gain if we have a maximization problem like INDEPEN-
DENT SET, or the minimum cost if we are dealing with a minimization problem such as the
TSP. For every instance I, let us denote by OPT(I) the value (benefit or cost) of the optimum
solution. It makes the math a little simpler (and is not too far from the truth) to assume that
OPT(I) is always a positive integer.
We have already seen an example of a (famous) approximation algorithm in Section 5.4:
the greedy scheme for SET COVER. For any instance I of size n, we showed that this greedy
algorithm is guaranteed to quickly find a set cover of cardinality at most OPT(I) log n. This
log n factor is known as the approximation guarantee of the algorithm.
More generally, consider any minimization problem. Suppose now that we have an algorithm
A for our problem which, given an instance I, returns a solution with value A(I). The
approximation ratio of algorithm A is defined to be
α A = max
I
A(I)
OPT(I) .
In other words, α A measures by the factor by which the output of algorithm A exceeds the
optimal solution, on the worst-case input. The approximation ratio can also be defined for
maximization problems, such as INDEPENDENT SET, in the same way—except that to get a
number larger than 1 we take the reciprocal.
So, when faced with an NP-complete optimization problem, a reasonable goal is to look for
an approximation algorithm A whose α A is as small as possible. But this kind of guarantee
might seem a little puzzling: How can we come close to the optimum if we cannot determine
the optimum? Let’s look at a simple example.
9.2.1 Vertex cover
We already know the VERTEX COVER problem is NP-hard.
VERTEX COVER
Input: An undirected graph G = (V, E).
Output: A subset of the vertices S ⊆ V that touches every edge.
Goal: Minimize |S|.
See Figure 9.3 for an example.
Since VERTEX COVER is a special case of SET COVER, we know from Chapter 5 that it can
be approximated within a factor of O(log n) by the greedy algorithm: repeatedly delete the
vertex of highest degree and include it in the vertex cover. And there are graphs on which the
greedy algorithm returns a vertex cover that is indeed log n times the optimum.
A better approximation algorithm for VERTEX COVER is based on the notion of a matching,
a subset of edges that have no vertices in common (Figure 9.4). A matching is maximal if no
more edges can be added to it. Maximal matchings will help us find good vertex covers, and
moreover, they are easy to generate: repeatedly pick edges that are disjoint from the ones
chosen already, until this is no longer possible.
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